ANALYSIS OF THE SINGULARITIES INFLUENCE ON THE FORWARD KINEMATICS SOLUTION AND THE GEOMETRY OF THE WORKSPACE OF THE GOUGH-STEWART PLATFORM
Abstract
One of the obligatory requirements for parallel mechanisms design is the exclusion from the
workspace of singularities in which the mechanism loses its controllability and malfunctions may
occur. The analysis of the workspace of the mechanisms of a parallel structure is more complicated
than that for the mechanisms of a serial structure, especially if the mechanism has more than
three degrees of freedom. The article considers the problem of analyzing the influence of singularities
on the solution of the forward kinematics and the geometry of the workspace 3/6 of the
Gough-Stewart platform (commercial name - "Hexapod"). A numerical algorithm for solving the
forward kinematics of platform has been developed. It is based on the direct use of the system of
equations of the platform's kinematic constraints. Approximation of the set of solutions to the system
of equations is based on deterministic methods of global optimization. An analysis of the
change in the number of forward kinematics near the zone of singularities is performed. The analysis
consists of two stages. The first stage consists in solving the forward kinematics for the position
and orientation of the platform, at which singularities arises. The second stage consists in
solving the forward kinematics for the case of a singularity and the case near a singularity.
As a result of solving the forward kinematics, a different number of forward kinematics solutions
for different cases was revealed. An algorithm has been synthesized that makes it possible to determine
a singularity-free workspace free for given ranges of change in the platform orientation
angles specified by Euler angles. An analysis of the dependence of the change in the volume of the
workspace depending on the range of change in the angles of the platform orientation was carried
out. The algorithms are implemented programmatically in the C++ programming language.
The modeling was performed using parallel computing and the implementation of the export of
three-dimensional models of the positions of the platform and workspace to the universal format of
three-dimensional models STL.
References
B.V., Krukhmalev V.A., Medvedeva T.N. Proektirovanie robotov i robototekhnicheskikh sistem
[Design of robots and robotic systems]. Rostov-on-Don, 2014, 196 p.
2. Kalyaev I.A., Lokhin V.M., Makarov I.M., Man'ko S.V., Romanov M.P., Yurevich E.I.
Intellektual'nye roboty [Intelligent robots]. Moscow, 2007, 360 p.
3. Kalyaev I.A., Gayduk A.R., Kapustyan S.G. Modeli i algoritmy kollektivnogo upravleniya v
gruppakh robotov [Models and algorithms of collective control in groups of robots]. Moscow,
2009, 280 p.
4. Glazunov V.A., Esina M.G., Bykov R.E. Upravlenie mekhanizmami parallel'noy struktury pri
perekhode cherez osobye polozheniya [Control of the mechanisms of a parallel structure when
passing through special provisions], Problemy mashinostroeniya i nadezhnosti mashin [Journal
of Machinery Manufacture and Reliability], 2004, No. 2, pp. 78-84.
5. Glazunov V.A., Chunikhin A.D. Razvitie issledovaniy mekhanizmov parallel'noy struktury
[Development of research into the mechanisms of a parallel structure], Problemy
mashinostroeniya i nadezhnosti mashin [Journal of Machinery Manufacture and Reliability],
2014, No. 3, pp. 37-43.
6. Merlet J.-P. Parallel Robots. Kluwer Academic Publishers, 2000, 355 p.
7. Kong X, Gosselin C.M. A class of 3-dof translational parallel manipulator with inear input/
output equations, Workshop on Fundamental Issues and Future Research for Parallel
Mechanisms and Manipulators, Québec City, Québec, Canada – 2002, pp. 25-32.
8. Craig J.J. Introduction to Robotics: Mechanics and Control. Stanford University: Stanford,
CA, USA, 2018, pp. 105-108.
9. Gosselin C.M., Angeles J. Singularity analysis of closed-loop kinematic chains, Transactions
on Robotics and Automatics, IEEE, 1990, Vol. 6 (3), pp. 281-290.
10. Gosselin C.M., Wang J. Singularity loci of planar parallel manipulators, Ninth World Congress
on the Theory of Machines and Mechanisms, Milano, Italy, 1995, pp. 1982-1986.
11. Hunt K.H. Kinematic Geometry of Mechanisms. Oxford, UK: Oxford University Press, 1978,
465 p.
12. Fichter E.F. A stewart platform-based manipulator: General theory and practical construction,
The International Journal of Robotics Research, 1986, Vol. 5 (2), pp. 157-182.
13. Merlet J.-P. Parallel manipulators part 2: Singular configurations and grassmann geometry,
Technical report, INRIA, Sophia Antipolis, France, 1989, Vol. 8 (5), pp. 45-56.
14. Merlet J.-P. Singular configurations of parallel manipulators and Grassmann geometry, The
International Journal of Robotics Research, 1989, Vol. 8 (5), pp. 45-56.
15. Dönmez D., Akçalı I.D., Avşar E., Aydın A., Mutlu H. Determination of particular singular
configurations of Stewart platform type of fixator by the stereographic projection method, Inverse
Problems in Science and Engineering, 2021, Vol. 29 (13), pp. 2925-2943.
16. Slavutin M., Sheffer A., Shai O. A Complete Geometric Singular Characterization of the 6/6
Stewart Platform, Journal of Mechanisms and Robotics, 2018, Vol. 10 (4).
17. Merlet J.-P. Interval Analysis and Robotics, Tracts in Advanced Robotics, 2010, Vol. 66,
pp. 147-156.
18. Malyshev D., Posypkin M., Rybak L., Usov A. Approaches to the determination of the working
area of parallel robots and the analysis of their geometric characteristics, Engineering Transactions,
2019, Vol. 67, No. 3, pp. 333-345.
19. Rybak L.A., Gaponenko E.V., Malyshev D.I., Virabyan L.G. The algorithm for planning the
trajectory of the 3-RPR robot, taking into account the singularity zones based on the method of
non-uniform covering, IOP Conference Series: Materials Science and Engineering, 2019,
Vol. 489, No. 1, 012060.
20. Rybak L., Malyshev D., Gaponenko E. Optimization Algorithm for Approximating the Solutions
Set of Nonlinear Inequalities Systems in the Problem of Determining the Robot Workspace,
Communications in Computer and Information Science, 2020, Vol. 1340, pp. 27-37.
